On the Ricci flow on Trees

2025/09/26 by Shuliang Bai, Bai, Shuliang, Bobo Hua +5
Mathematics · Physics and Astronomy · #53C21 #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2509.22140

openalex publication_date 2025/09/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we study the evolution of metrics on finite trees under continuous-time Ricci flows based on the Lin-Lu-Yau version of Ollivier Ricci curvature. We analyze long-time dynamics of edge weights and curvatures, providing precise characterizations of their limiting behaviors. We prove that the Ricci flow converges to metric with zero curvature on edges whose normalized weights converge to positive values only if the tree is a caterpillar tree.

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Related

• Ollivier Ricci-flow on weighted graphs
• A version of Bakry-Émery Ricci flow on a finite graph
• Convergence stability for Ricci flow on manifolds with bounded geometry
• A Ricci flow on graphs from effective resistance
• Discrete Ollivier-Ricci curvature